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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<b>thus </b><a NAME="E1:10"/>
the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:10"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a'</font>,<font color="Olive">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:10_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 ;<br/>

<a NAME="E2:10_2"/><b>then </b><i><font color="Green">E7</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:10_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E4:10_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:10"/>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><span class="p3"><a href="cat_2.html#K13">[</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span><a href="cat_2.html#K13">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">d</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E4:10"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E5:10"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:10"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
